Okay. So for this problem were given a density function as of tea and were asked to find the expectation of tea and standard deviation to start out. Well, look at what we need to do to find expectation. We're gonna integrate over our domain, which it's not written here, but as 13.4 up to 62. So we're gonna integrate from 13.4 to 62 this entire function, Uh, s of T times TV. Uh, DT. And so what? This is going to be since this is just a polynomial multiplying it by t is just gonna raise all of our powers by one. So we're gonna get this whole constant times five this one That's four times three attempts to our attempts t to the to t. Uh, and this last one will be 1 37.5 times tea, right? We're just multiplying t through this whole thing. Um, and we can move the constant part upside. So integrating this is a little bit annoying, but not too hard, Since all of the terms look like some form of a constant times are variable t to some power When we integrate this with respect to t We're just going to get see over plus one, do you, k plus one, right? So we just need toe up the power by one and divide the constant front by that. So to save us some space, I'm gonna go ahead and ah, just write out what this integral evaluates out to. So there's a lot of different things online that you can use to solve that or to check your work, But it would be just a little bit tedious to do manually. So our answer is gonna come out to be 31 0.741 and that's gonna be just our first part. So after we've got expectation, the next step in this problem is finding out, uh, the standard deviation. And so if we want standard deviation of X, what we really want is a squared off the variants, and we know that the variants is equal to expected t squared minus expected t squared. And so we already have expected t. It's this 31.7491 and we can easily square it. So what we need to do is find this expected t squared term expected t squared. And so what? That's gonna be very similar, Integral. It's going to be our density function S of T. But this time it's times t squared because we're trying to find expectation of T Square and again, all since this is a polynomial, all that we're gonna do is be using this formula again, right? And so to save a little bit of time, we'll just write down. But the results this from that we're gonna get 1141.5. And so this is not quite where we want to stop for the second part. This is just are expected t squared from that 1000. Ah, 141.5. We need to subtract off 31.74 91 squared. So we need to take This is 31.74 91 squared as that's that's are expected, t um, doing that will give us are variants. And so if you plug this into a calculator, you'll see that the variances that equal to 1 33.4946 and to get to the standard deviation then we want the square root of 1 33.4946 and so bad comes out to be oops, about 11.55 and so that will be our final answer for the second part of this question.